1. **Problem Statement:** Given a 3x3 matrix $B$ such that $B^2 = B$, determine which statements about $B$ must be true. 2. **Key Property:** The equation $B^2 = B$ means $B$ is idempotent. 3. **Important facts about idempotent matrices:** - Eigenvalues of $B$ are either 0 or 1. - $B$ is diagonalizable with eigenvalues 0 or 1. 4. **Analyze each statement:** **A. $B$ is invertible:** - If $B$ were invertible, then $B^2 = B$ implies $B = I$ (identity matrix). - So $B$ could be invertible only if $B = I$. - But $B$ could also be a projection matrix (not invertible). - Therefore, $B$ is not necessarily invertible. **B. $ ext{det}(B) = 0$:** - Since eigenvalues are 0 or 1, determinant is product of eigenvalues. - If any eigenvalue is 0, determinant is 0. - But if $B = I$, all eigenvalues are 1, determinant is 1. - So determinant can be 0 or 1, not necessarily 0. **C. $ ext{det}(B^2) = ext{det}(B)$:** - Using property of determinants: $ ext{det}(B^2) = ( ext{det}(B))^2$. - Given $B^2 = B$, so $ ext{det}(B^2) = ext{det}(B)$. - Thus, $( ext{det}(B))^2 = ext{det}(B)$. - This implies $ ext{det}(B)( ext{det}(B) - 1) = 0$. - So $ ext{det}(B) = 0$ or $ ext{det}(B) = 1$. - This is always true for idempotent $B$. **D. $ ext{det}(B) = - ext{det}(B)$:** - This implies $ ext{det}(B) = 0$. - But as shown, determinant can be 1 as well. - So this is not necessarily true. 5. **Conclusion:** Only statement C must be true for any idempotent matrix $B$. **Final answer:** C